A Mathematical Gift I: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita PDF

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

ISBN-10: 0821832824

ISBN-13: 9780821832820

This booklet will convey the sweetness and enjoyable of arithmetic to the study room. It bargains critical arithmetic in a full of life, reader-friendly sort. incorporated are workouts and plenty of figures illustrating the most techniques.
The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors talk about the Poincaré-Hopf theorem on severe issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses quite a few facets of the idea that of size, together with the Peano curve and the Poincaré method. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it really is proved that the 3-dimensional sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a chain of lectures given via the authors at Kyoto collage (Japan).

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Extra resources for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)

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J:sinn-281 ... Sin8n-2d81 ... d8n-l n-l J . 3) 2j1/2 j-l n = T(n/2)" These computations are straightforward. 1. ~ dO'n-l (e) (xe IRn). o over Sn-l' When n = 1, So = {-1,1}, is the measure consisting of a point mass of 1/2 at each of -1 and 1 and (xelR). 22(X) =! 2n is a radial function, (xe IR"). 5) This is a consequence of the fact that O'n-l is invariant under isometries of IR n which leave the origin fixed. With this observation we put (r ~ 0). \ Is ... 7) §7. 8) ~O). 9) . 2. oiru) da(u) (0 :s; r < (0).

1. 4). Proof. Our opening remarks establish the conclusion in case 2 < p ~ 00. §8. 12). 9). To show that every function in N(U) (0 < p ~ 2) has the indicated form we need the following lemma: Consider the sequence qJn(r) = 1 00 1_e-ru u o (0 ~ r) dPn(u) where Pn is a positive measure on IR+ and (n= 1,2, ... ). If cpir) ---+ cp(r) for each re IR+ and cp is continuous at zero, then cp(r) = 1_e- ru dP(u) o u frO (0 ~ r) where P is a positive measure on IR+ such that roo J1 dP(u) < 00. u The sequenceYn of positive measures on [0,00 ] defined by dYn(u) = 1- e- U dPn(u) u is uniformly bounded since J~ dYn(u) = CPn(1) ---+ cp(1) as n ---+ 00.

49 §11. 4). Pick an index Oel and define Bj = B(Yo,ro)nB(Y/,rj) (iel). Each of the weakly closed sets B/ is contained in the weakly compact ball B(yo,ro). 5) will follow provided {B(Ybr/): iel} possesses the finite intersection property. To this end choose finitely many balls indexed i = 1, ... ,m. 2) We may assume x =1= x/ (1 S; is; m) for if x = Define the real function J on H 1 SiSm IItll-+oo then Y = yje by J(t) = max Since J is continuous, lim J(t) = Xj m n B(y/,r/). /=1 Ilt-YIII ... IIX-Xtil + 00 and J(t) 2!

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A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19) by Kenji Ueno, Koji Shiga, Shigeyuki Morita

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